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High School Math : Exponents

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #4 : Understanding Asymptotes

$y=\frac{3x^2-5}{2x^2+7}$

What are the horizontal asymptotes of this equation?

Possible Answers:

There are no horizontal asymptotes.

y=0

$y=\frac{2}{3}$

$y=\frac{3}{2}$

y=1

Correct answer:

$y=\frac{3}{2}$

Explanation:

Since the exponents of the variables in both the numerator and denominator are equal, the horizontal asymptote will be the coefficient of the numerator's variable divided by the coefficient of the denominator's variable.

For this problem, since we have $\frac{3x^2}{2x^2}$ , our asymptote will be $y=\frac{3}{2}$ .

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Example Question #5 : Understanding Asymptotes

$y=\frac{3x^2-5}{2x^2+7}$

What are the vertical asymptotes of the equation?

Possible Answers:

$x=\frac{7}{2}$

$x=-\sqrt{\frac{7}{2}}$

There are no vertical asymptotes.

$x=-\sqrt{\frac{7}{2}}, \sqrt{\frac{7}{2}}$

There are no real vertical asymptotes.

Correct answer:

There are no real vertical asymptotes.

Explanation:

To find the vertical asymptotes, we set the denominator equal to zero and solve.

$0={2x^2+7}$

-7=2x^2

$\frac{-7}{2}=x^2$

$\sqrt{-\frac{7}{2}}=\sqrt{x^2}$

$\sqrt{-\frac{7}{2}}=x$

Since we'd be trying to find a negative number, we have no real solution. Therefore, there are no real vertical asymptotes.

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Example Question #4 : Solving And Graphing Exponential Equations

$y=\frac{x^2-64}{x+2}$

What are the vertical asymptotes of this equation?

Possible Answers:

x=0

x=2,-2

x=8,-8

x=-2

There are no real vertical asymptotes for this function.

Correct answer:

x=-2

Explanation:

To find the vertical asymptotes, we set the denominator equal to zero.

0=x+2

-2=x

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Example Question #1 : Understanding Asymptotes

$y=\frac{x^2-9}{x^2-16}$

What is the horizontal asymptote of this equation?

Possible Answers:

y=1

y=0

y=3

y=2

There is no horizontal asymptote.

Correct answer:

y=1

Explanation:

Look at the exponents of the variables. Both our numerator and denominator are 1x^2 . Therefore the horizontal asymptote is calculated by dividing the coefficient of the numerator by the coefficient of the denomenator.

$y=\frac{1}{1}=1$

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Example Question #1 : Understanding Asymptotes

Find the vertical asymptote(s) of $y=\frac{x^2+9x+8}{x^2-44}$ .

Possible Answers:

$x=2\sqrt{11}$

There are no real vertical asymptotes for this function.

$x=2\sqrt{11}$ and $x=-2\sqrt{11}$

x=1

x=-1 and x=-8

Correct answer:

$x=2\sqrt{11}$ and $x=-2\sqrt{11}$

Explanation:

To find the vertical asymptotes, we set the denominator of the fraction equal to zero, as dividing anything by zero is undefined.

Take our given equation, $y=\frac{x^2+9x+8}{x^2-44}$ , and now set the denominator equal to zero:

0=x^2-44

44=x^2

$\sqrt{44}=\sqrt{x^2}$

is not a perfect square, but let's see if we can pull anything out.

$\sqrt{4*11}=\sqrt{x^2}$

$\sqrt{4}*\sqrt{11}=\sqrt{x^2}$

$2\sqrt{11}=x$

Don't forget that there is a negative result as well:

$x=-2\sqrt{11}$ .

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Example Question #115 : Algebra Ii

Find the vertical asymptote(s) of $y=\frac{x^3+2x+8}{x^2-36}$ .

Possible Answers:

x=-6 and x=6

x=36

x=6

There are no real vertical asymptotes.

$x=\frac{3}{2}$

Correct answer:

x=-6 and x=6

Explanation:

To find the vertical asymptotes, we set the denominator of the fraction equal to zero, as dividing anything by zero is "undefined." Since it's undefined, there's no way for us to graph that point!

Take our given equation, $y=\frac{x^3+2x+8}{x^2-36}$ , and now set the denominator equal to zero:

0=x^2-36 .

36=x^2

$\sqrt{36}=\sqrt{x^2}$

Don't forget, the root of a positive number can be both positive or negative ( -6^2=36 as does 6^2 ), so our answer will be $\pm6=x$ .

Therefore the vertical asymptotes are at x=-6 and x=6 .

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Example Question #21 : Exponents

Find the horizontal asymptote(s) of $y=\frac{x^3+2x+8}{x^2-36}$ .

Possible Answers:

There are no real horizontal asymptotes.

y=0

$y=\frac{3}{2}$

y=1

y=6 and y=-6

Correct answer:

There are no real horizontal asymptotes.

Explanation:

To find the horizontal asymptote of the function, look at the variable with the highest exponent. In the case of our equation, $y=\frac{x^3+2x+8}{x^2-36}$ , the highest exponent is x^3 in the numerator.

When the variable with the highest exponent is in the numberator, there are NO horizontal asymptotes. Horizontal asymptotes only appear when the greatest exponent is in the denominator OR when the exponents have same power in both the denominator and numerator.

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Example Question #1 : Quadratic Functions

What are the -intercepts of the equation?

$y=\frac{x^2-9}{x^2-16}$

Possible Answers:

x=4,-4

x=3

There are no -intercepts.

x=3,-3

x=2

Correct answer:

x=3,-3

Explanation:

To find the x-intercepts of the equation, we set the numerator equal to zero.

0=x^2-9

9=x^2

$\sqrt{9}=\sqrt{x^2}$

3,-3=x

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Example Question #1 : Solving Exponential Functions

Find the vertical asymptote of the equation.

$y=\frac{x^2-9}{x^2-16}$

Possible Answers:

x=4, -4

x=4

There are no vertical asymptotes.

x=1

x=0

Correct answer:

x=4, -4

Explanation:

To find the vertical asymptotes, we set the denominator of the function equal to zero and solve.

0=x^2-16

16=x^2

$\sqrt{16}=\sqrt{x^2}$

4, -4=x

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Example Question #11 : Understanding Asymptotes

$y=\frac{x^2-64}{x+2}$

What is the horizontal asymptote of this equation?

Possible Answers:

y=-2

y=1

There is no horizontal asymptote.

y=32

y=0

Correct answer:

There is no horizontal asymptote.

Explanation:

Since the exponent of the leading term in the numerator is greater than the exponent of the leading term in the denominator, there is no horizontal asymptote.

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High School Math : Exponents

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #4 : Understanding Asymptotes

Example Question #5 : Understanding Asymptotes

Example Question #4 : Solving And Graphing Exponential Equations

Example Question #1 : Understanding Asymptotes

Example Question #1 : Understanding Asymptotes

Example Question #115 : Algebra Ii

Example Question #21 : Exponents

Example Question #1 : Quadratic Functions

Example Question #1 : Solving Exponential Functions

Example Question #11 : Understanding Asymptotes

All High School Math Resources

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